5. The decisive small-parameter calculation

The next phase no longer searched over unrelated formulas. We fixed

\[ \tag{5.1}\label{eq:pure-power-candidate} P(x)=\prod_{j=1}^m L_j^{2/m}, \qquad L_j=1-\ip{x}{p_j}, \]

and asked whether this balanced power was always viable away from its zeros when \(k\) was sufficiently small.

5.1 The small-\(k\) first-order expansion

Let \(S=1-\nu_1^{2}-\cdots-\nu_\ell^{2}\). At a fixed point with \(S\gt{}0\), direct expansion of (3.3) and (3.6) gives

\[ \tag{5.2}\label{eq:small-k-B-expansion} W=1+kw, \qquad w=P-\frac{1}{2}\sum_{i=1}^{\ell}\nu_iP_i, \qquad B_{aa}=1+k\mu_a+O(k^2), \]

where

\[ \mu_a =-\frac{1}{2}\sum_{i,j=1}^{\ell} P_{ij}\ip{e_i}{\tau_a}\ip{e_j}{\tau_a} -\frac{ \left(\sum_i\nu_i\ip{e_i}{\tau_a}\right) \left(\sum_jP_j\ip{e_j}{\tau_a}\right) }{S} -\frac{ P\left(\sum_i\nu_i\ip{e_i}{\tau_a}\right)^2 }{S^2}. \]

These were precisely the expressions implemented in the first-order numerical code: the input was \(P\), its first two derivatives, the point \(x\), and the frame coefficients \(\ip{e_i}{\tau_a}\).

Put

\[ p_0(t)=\prod_{a=1}^{\ell}(t-\mu_a). \]

Substitution of (5.2) into the threshold polynomial (3.7) shows that

\[ \tag{5.3}\label{eq:critical-root} s=1+k\rho_++O(k^2), \qquad \rho_+\text{ is the largest real root of }p_0'. \]

The ambient dimension \(n\) disappears from this first-order root; only the active rank \(\ell\) remains. Define the gap coefficient

\[ \varrho=w-\rho_+. \]

Then

\[ \tag{5.4}\label{eq:normalized-small-k-threshold} W-s=k\varrho+O(k^2), \qquad \frac{s}{W}=1-k\varrho+O(k^2). \]

Consequently, the negative first-order term occurs in the normalized ratio \(s/W\), whereas the unnormalized threshold in (5.3) has the positive first-order expansion. The clean sufficient condition is

\[ \class{key-formula-math}{\varrho\gt{}0.} \]

5.2 The first-order criteria in ranks two and three

For \(\ell=2\), equation (5.3) immediately gives

\[ \rho_+=\frac{\mu_1+\mu_2}{2}, \qquad \varrho=w-\frac{\mu_1+\mu_2}{2}. \]

Equivalently, the first-order test is simply

\[ 2W-(B_{11}+B_{22})=2k\varrho+O(k^2)\gt{}0. \]

For the balanced product (5.1), direct substitution and a Cauchy–Schwarz estimate give \(\varrho\gt{}0\) away from the zeros. Thus the balanced exponent \(2/m\) passes the rank-two first-order test.

For \(\ell=3\), put

\[ \sigma_1=\mu_1+\mu_2+\mu_3, \qquad \sigma_2=\mu_1\mu_2+\mu_1\mu_3+\mu_2\mu_3. \]

The critical-root equation becomes

\[ \tag{5.5}\label{eq:l3-rho-equation} 3\rho^2-2\sigma_1\rho+\sigma_2=0, \qquad \rho_+=\frac{\sigma_1+\sqrt{\sigma_1^2-3\sigma_2}}{3}. \]

Accordingly, \(\varrho\gt{}0\) is equivalent to placing \(w\) to the right of the larger root. A convenient pair of scalar checks is

\[ \tag{5.6}\label{eq:l3-sigma-test} 3w-\sigma_1\gt{}0, \qquad 3w^2-2\sigma_1w+\sigma_2\gt{}0. \]

This was the form used in the rank-three tests.

5.3 Rank-three scans and analytic confirmation

The numerical work tested (5.6) for coordinate, symmetric, clustered, tilted, and random prescribed directions. Every entry in Table 5.1 concerns the first-order gap

\[ \varrho =\left.\partial_k(W-s)\right|_{k=0^+} =w-\rho_+. \]

For each listed configuration, we numerically evaluated the normalized boundary gap \(\varrho/P\) and the first-order gap \(\varrho\). The randomized rows record separate scans for \(m=2,\ldots,5\).

Pole configuration \(m\) \(2/m\) Boundary \(\min(\varrho/P)\) Sampled \(\min\varrho\)
Coordinate triple 3 \(2/3\) \(0.5006385\) \(0.0999129\)
Tetrahedral four 4 \(1/2\) \(0.5000182\) \(0.3344756\)
Octahedral six 6 \(1/3\) \(0.5000031\) \(0.3157721\)
Random four 4 \(1/2\) \(0.5009658\) \(0.0426013\)
Random six 6 \(1/3\) \(0.5000017\) \(0.0427318\)
Random eight 8 \(1/4\) \(0.5010803\) \(0.1085162\)
Randomized two-pole (60 tries) 2 \(1\) --- \(2.41\times10^{-1}\)
Randomized three-pole (60 tries) 3 \(2/3\) --- \(2.06\times10^{-1}\)
Randomized four-pole (60 tries) 4 \(1/2\) --- \(2.28\times10^{-1}\)
Randomized five-pole (60 tries) 5 \(2/5\) --- \(2.92\times10^{-1}\)
Table 5.1. Rank-three first-order scans for the balanced power product.

All four randomized families passed the first-order scans. The values in the table were numerical evidence only, but together they supported the stability of the balanced \(2/m\) product away from the zero set. The computation also showed why samples taken too close to a zero could be misleading at fixed \(k\): the first-order expansion is not uniform there.

After these scans, a detailed analytic calculation reduced (5.6) to a covariance estimate. It showed that \(w\) lies strictly to the right of the larger root in (5.5) for every set of distinct prescribed directions. Consequently, \(\varrho\) has a positive minimum on every compact set separated from the zeros. Continuity and (5.4) then give

\[ W-s\gt{}0 \]

there whenever \(k\gt{}0\) is chosen sufficiently small.

This completed the away-from-zero part for \(\ell\leq3\), but it did not solve the original problem. Near a prescribed direction,

\[ P(x)\sim C_jt^{2/m}, \qquad t=1-\ip{x}{p_j}, \qquad C_j\gt{}0, \]

and for every fixed positive \(k\) the uniform lower bound could still fail in a thin neighborhood of that zero. Decreasing \(k\) shrank the bad region but never removed it. The remaining task was therefore sharply defined: keep the successful \(2/m\) tail away from the zeros and replace its local vanishing model. This led to the linear calculation in the next section.

Gaoming Wang
Gaoming Wang
Assistant Professor

My research interests include Geometric Analysis and Partial Differential Equations.